Equality of directional group orders for two-step rules

Let $\mathcal Q$ be the class of two-step rules, and let $G_e,G_n,G_w,G_s$ denote the corresponding groups for the four directions. Directional group-order conjecture. For every two-step rule in $\mathcal Q$,

$$|G_e|=|G_n|=|G_w|=|G_s|.$$

The claim is based on computational experiments over the quarter-plane models; the supplied source gives no proof or resolution, so it remains open.